Got this problem today. Nearly got mad about this. I tried everything mentioned above. I visited the evernote.com website and logged in. After I closed the site and and reopened it, I had to login again.
I now discovered, that suddenly a "remember me for 30 days" option was available on the login page. I clicked this and now the web clipper works again.
I'm absolutely sure, that this option was not visible to me the whole day. Strange behavior.